Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  felapton GIF version

Theorem felapton 2120
 Description: "Felapton", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜑 is 𝜒, and some 𝜑 exist, therefore some 𝜒 is not 𝜓. (In Aristotelian notation, EAO-3: MeP and MaS therefore SoP.) For example, "No flowers are animals" and "All flowers are plants", therefore "Some plants are not animals". (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
felapton.maj 𝑥(𝜑 → ¬ 𝜓)
felapton.min 𝑥(𝜑𝜒)
felapton.e 𝑥𝜑
Assertion
Ref Expression
felapton 𝑥(𝜒 ∧ ¬ 𝜓)

Proof of Theorem felapton
StepHypRef Expression
1 felapton.e . 2 𝑥𝜑
2 felapton.min . . . 4 𝑥(𝜑𝜒)
32spi 1516 . . 3 (𝜑𝜒)
4 felapton.maj . . . 4 𝑥(𝜑 → ¬ 𝜓)
54spi 1516 . . 3 (𝜑 → ¬ 𝜓)
63, 5jca 304 . 2 (𝜑 → (𝜒 ∧ ¬ 𝜓))
71, 6eximii 1582 1 𝑥(𝜒 ∧ ¬ 𝜓)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  ∀wal 1333  ∃wex 1472 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-ial 1514 This theorem depends on definitions:  df-bi 116 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator